Source: Adv. in Appl. Probab.
Volume 40, Number 1
We prove that the Heston volatility is Malliavin differentiable
under the classical Novikov condition and give an explicit
expression for the derivative. This result guarantees the
applicability of Malliavin calculus in the framework of the Heston
stochastic volatility model. Furthermore, we derive conditions on
the parameters which assure the existence of the second Malliavin
derivative of the Heston volatility. This allows us to apply
recent results of Alòs (2006) in order to derive
approximate option pricing formulae in the context of the Heston
model. Numerical results are given.
Full-text: Access denied (no subscription
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Alòs, E. (2006). An extension of the Hull and White formula with applications to option pricing approximation. Finance Stoch. 10, 353--365.
Alòs, E. and Nualart, D. (1998). An extension of Itô's formula for anticipating processes. J. Theoret. Prob. 11, 493--514.
Alòs, E., Leon, J. A. and Vives, J. (2007). On the shorttime behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571--598.
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion---Facts and Formulae, 2nd edn. Birkhäuser, Basel.
Bossy, M. D. A. (2004). An efficient discretization scheme for one dimensional SDE's with a diffusion coefficient function of the form $\vert x \vert^\alpha$, $\alpha \in [1/2,1)$. Res. Rep. 5396 INRIA.
Detemple, J., Garcia, R. and Rindisbacher, M. (2005). Representation formulas for Malliavin derivatives of diffusion processes. Finance Stoch. 9, 349--367.
Geman, H. and Yor, M. (1993). Bessel Processes, Asian Options and perpetuities. Math. Finance 3, 349--375.
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327--343.
Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42, 281--300.
Jeanblanc, M. (2006). Cours de calcul stochastique. Master 2IF EVRY. Lecture Notes, University of Évry. Available at http://www.maths.univ-evry.fr/pages$\_$perso/jeanblanc/.
Karatzas, I. and Shreve, S.-E. (1988). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113). Springer, New York.
Mathematical Reviews (MathSciNet): MR917065
Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
Yor, M. (1992). Some Aspects of Brownian Motion, Part 1, Some Special Functionals. Birkhäuser, Basel.